A product-estimate for Ginzburg–Landau and corollaries

We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension. It allows to retrieve existing lower bounds on the energy, to extend them to the case of unbounded vorticity, and to get a few other corollaries. It also provides a new estimate on the time-variation for time-dependent families, which has applications for the study of Ginzburg–Landau dynamics.